| L(s) = 1 | + 18.1·2-s + 27·3-s + 202.·4-s + 103.·5-s + 490.·6-s + 1.34e3·8-s + 729·9-s + 1.87e3·10-s + 1.94e3·11-s + 5.45e3·12-s + 9.44e3·13-s + 2.79e3·15-s − 1.38e3·16-s − 1.35e4·17-s + 1.32e4·18-s + 5.23e4·19-s + 2.09e4·20-s + 3.53e4·22-s + 6.18e4·23-s + 3.63e4·24-s − 6.74e4·25-s + 1.71e5·26-s + 1.96e4·27-s − 3.99e4·29-s + 5.07e4·30-s + 3.12e5·31-s − 1.97e5·32-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 0.577·3-s + 1.57·4-s + 0.369·5-s + 0.927·6-s + 0.930·8-s + 0.333·9-s + 0.594·10-s + 0.440·11-s + 0.911·12-s + 1.19·13-s + 0.213·15-s − 0.0847·16-s − 0.666·17-s + 0.535·18-s + 1.75·19-s + 0.584·20-s + 0.707·22-s + 1.05·23-s + 0.537·24-s − 0.863·25-s + 1.91·26-s + 0.192·27-s − 0.304·29-s + 0.343·30-s + 1.88·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.508783370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.508783370\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 18.1T + 128T^{2} \) |
| 5 | \( 1 - 103.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 1.94e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.44e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.35e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.18e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.99e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.12e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.25e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.37e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.19e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.33e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.88e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.45e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.82e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.98e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.78e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.26e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.65e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.98e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.14e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.58e6T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.93865351216336851437853565239, −11.13833648159016376307251474270, −9.703302595590740108527783428528, −8.575253933002246912810242331253, −7.07682870669022822684609483597, −6.09948457380600316107967189598, −4.98318185228013321237481703565, −3.78113670665006230851354301454, −2.88488718855232940085160632313, −1.43531620972989278075257021577,
1.43531620972989278075257021577, 2.88488718855232940085160632313, 3.78113670665006230851354301454, 4.98318185228013321237481703565, 6.09948457380600316107967189598, 7.07682870669022822684609483597, 8.575253933002246912810242331253, 9.703302595590740108527783428528, 11.13833648159016376307251474270, 11.93865351216336851437853565239
